Abstract
In the present work, we introduce the subclass T k
γ,α(ϕ), of starlike
functions with respect to k-symmetric points of complex order γ (γ ̸= 0)
in the open unit disc △. Some interesting subordination criteria, inclusion
relations and the integral representation for functions belonging to this class
are provided. The results obtained generalize some known results, and some
other new results are obtained.
Results & Lemmas (10)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1
Lemma 1 ([10]). Let k, ϑ be complex numbers. Suppose that h(z) is convex and univalent in △with (1.8) h(0) = 1 and Re [kh(z) + ϑ] > 0 (z…
Lemma 1 ([10]). Let k, ϑ be complex numbers. Suppose that h(z) is convex and univalent in △with (1.8) h(0) = 1 and Re [kh(z) + ϑ] > 0 (z ∈△), and let q(z) be analytic in △with q(0) = 1 and q(z) ≺h(z). If p(z) = 1 + p1z + p2z2 + · · · ∈℘with p(0) = 1, then p(z) + zp′(z) kq(z) + ϑ ≺h(z) implies that p(z) ≺h(z).
Lemma 2
Lemma 2 (see [6, 7]). Let k, ϑ be complex numbers. Suppose that h(z) is convex and univalent in △and satisfies (1.8). If p(z) = 1+p1z +p2z2…
Lemma 2 (see [6, 7]). Let k, ϑ be complex numbers. Suppose that h(z) is convex and univalent in △and satisfies (1.8). If p(z) = 1+p1z +p2z2 +. . . ∈℘and satisfies the subordination p(z) + zp′(z) kp(z) + ϑ ≺h(z), then p(z) ≺h(z). 2. Main result. Unless otherwise mentioned, we assume throughout this article that f ∈A, α > 0, ϕ ∈℘and γ ∈C∗.
Proposition 1.
Proposition 1. Let f ∈Tγ,α(ϕ) and Re 1 α [αγ (ϕ(z) −1) + 1] > 0, then f ∈Sγ(ϕ).
Proposition 1. Let f ∈Tγ,α(ϕ) and Re 1 α [αγ (ϕ(z) −1) + 1] > 0, then f ∈Sγ(ϕ).
Proposition 2.
Proposition 2. Let Re 1 α [αγ (ϕ(z) −1) + 1] > 0. Then (2.2) F(z) = Iα(f) = 1 αz(1/α)−1 z Z 0 t(1/α)−2f(t)dt ∈Sγ(ϕ) whenever f(z) ∈Sγ(ϕ).
Proposition 2. Let Re 1 α [αγ (ϕ(z) −1) + 1] > 0. Then (2.2) F(z) = Iα(f) = 1 αz(1/α)−1 z Z 0 t(1/α)−2f(t)dt ∈Sγ(ϕ) whenever f(z) ∈Sγ(ϕ).
Theorem 1.
Theorem 1. Let f ∈T k γ,α(ϕ) and Re 1 α [1 + αγ (ϕ(z) −1)] > 0. Then fk defined by (1.2) is in Tγ,α(ϕ). Further, we have fk ∈Sγ(ϕ).
Theorem 1. Let f ∈T k γ,α(ϕ) and Re 1 α [1 + αγ (ϕ(z) −1)] > 0. Then fk defined by (1.2) is in Tγ,α(ϕ). Further, we have fk ∈Sγ(ϕ).
Theorem 2.
Theorem 2. Let f ∈T k γ,α(ϕ) and Re 1 α [1 + αγ [ϕ(z) −1]] > 0. Then f ∈ Sk γ(ϕ).
Theorem 2. Let f ∈T k γ,α(ϕ) and Re 1 α [1 + αγ [ϕ(z) −1]] > 0. Then f ∈ Sk γ(ϕ).
Theorem 3.
Theorem 3. Let f ∈Sk γ(ϕ) and Re 1 α [1 + αγ (ϕ(z) −1)] > 0 in △and let F be the integral operator defined by (2.2), then F ∈Sk γ(ϕ).
Theorem 3. Let f ∈Sk γ(ϕ) and Re 1 α [1 + αγ (ϕ(z) −1)] > 0 in △and let F be the integral operator defined by (2.2), then F ∈Sk γ(ϕ).
Theorem 4.
Theorem 4. Let Re 1 α [1 + αγ [ϕ(z) −1]] > 0. Then Fk γ,α(ϕ) ⊂Fk γ,0(ϕ). Now, we give the integral representations of functions belonging…
Theorem 4. Let Re 1 α [1 + αγ [ϕ(z) −1]] > 0. Then Fk γ,α(ϕ) ⊂Fk γ,0(ϕ). Now, we give the integral representations of functions belonging to the classes T k γ,α(ϕ).
Theorem 5.
Theorem 5. Let f ∈T k γ,α(ϕ), then we have (2.11) fk(z) = 1 αz1−1 α Z z 0 exp γ k k−1
Theorem 5. Let f ∈T k γ,α(ϕ), then we have (2.11) fk(z) = 1 αz1−1 α Z z 0 exp γ k k−1
Theorem 5.
Theorem 5. □
Theorem 5. □
Function classes studied:
Related Papers